The Most Unstable Conditions of Modulation Instability

Modulation instability is one of the most ubiquitous types of instabilities in nature. As one of the key characteristics of modulation instability, the most unstable condition attracts lots of attention. The most unstable condition is investigated here with two kinds of initial wave systems via a numerical high-order spectral method (HOS) for surface water wave field. Classically, one carrier wave and a pair of sidebands are implied as the first kind of initial wave system: “seeded†wave system. In the second kind of initial wave system: “un-seeded†wave system, only one carrier wave is implied. Two impressive new results are present. One result shows that the grow rates of lower and upper sideband are different within the “seeded†wave system. It means that, for a given wave steepness, the most unstable lower sideband is not in pair with the most unstable upper sideband. Another result shows the fastest growing sidebands are exactly in pair from “unseeded†wave system. And the most unstable conditions of “unseeded†wave system are more or less the mean value of those derived from the lower sidebands and upper sidebands within the “seeded†wave system.